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Approachability at the second successor of a singular cardinal. (English) Zbl 1184.03046

Summary: We prove that if \(\mu \) is a regular cardinal and \(\mathbb P\) is a \(\mu \)-centered forcing poset, then \(\mathbb P\) forces that \((I[\mu ^{++}])^{V}\) generates \(I[\mu ^{++}]\) modulo clubs. Using this result, we construct models in which the approachability property fails at the successor of a singular cardinal. We also construct models in which the properties of being internally club and internally approachable are distinct for sets of size the successor of a singular cardinal.

MSC:

03E05 Other combinatorial set theory
03E35 Consistency and independence results
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